##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory Seminar

## Brian Lawrence

## Sparsity of Integral Points on Moduli Spaces of Varieties

##### Abstract:

Interesting moduli spaces don't have many integral points. More precisely, if X is a variety over a number field, admitting a variation of Hodge structure whose associate period map is injective, then the number of S-integral points on X of height at most H grows more slowly than $H^ε$, for any positive ε. This is a sort of weak generalization of the Shafarevich conjecture; it is a consequence of a point-counting theorem of Broberg, and the largeness of the fundamental group of X. Joint with Ellenberg and Venkatesh.

### April 28, 2022

### 2:00 PM

Pre-talk at 1:20 PM

APM 6402 and Zoom

See https://www.math.ucsd.edu/~nts

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